**The seminar meets regularly on Wednesdays** at **4pm** in **Eckhart 202**. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Cornelia Mihaila.

More about the long time behavior of stochastic HJ equations

TBA

October 7

October 9

Regularization of some nonlinear PDE’s in Hilbert spaces

TBA

The Lax-Oleinik representation in non-compact setting

We will be interested in viscosity solutions of the evolution Hamilton-Jacobi equation ∂_tU + H(x, ∂_xU) = 0.

Here we think of the case where U : [0, +∞)×M → R, with M is a manifold. If M is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition U|{0}×M. This in turn implies the representation by a Lax-Oleinik type formula.

When M is not compact, the global maximum principle does not immediately hold. Hitoshi Ishii and his coworkers obtained results about 10 years ago under some restrictions when M = R^n. Basically the restrictions are about controlled growth at infinity.

We will explain that under the hypothesis that H is Tonelli, all continuous solutions of the evolution Hamilton-Jacobi equation above satisfy the Lax-Oleinik representation even for non-compact M. This of course will imply uniqueness for a given initial condition.

Moreover, we will also show that if any pointwise finite U is given by the Lax-Oleinik representation is automatically continuous and therefore a viscosity solution.

(Postponed) Front propagation in a nonlocal reaction-diffusion equation

We consider a reaction-diffusion equation with a nonlocal reaction term. This PDE arises as a model in evolutionary ecology. We study the regularity properties and asymptotic behavior of its solutions.

Large deviations for conservative, stochastic PDE and non-equilibrium fluctuations

Macroscopic fluctuation theory provides a general framework for far from equilibrium thermodynamics, based on a fundamental formula for large fluctuations around (local) equilibria. This fundamental postulate can be informally justified from the framework of fluctuating hydrodynamics, linking far from equilibrium behavior to zero-noise large deviations in conservative, stochastic PDE. In this talk, we will give rigorous justification to this relation in the special case of the zero range process. More precisely, we show that the rate function describing its large fluctuations is identical to the rate function appearing in zero noise large deviations to conservative stochastic PDE, by means of proving the Gamma-convergence of rate functions to approximating stochastic PDE. The proof of Gamma-convergence is based on the well-posedness of the skeleton equation -- a degenerate parabolic-hyperbolic PDE with irregular coefficients, the proof of which extends DiPerna-Lions' renormalization techniques to nonlinear PDE.

Optimal boundary regularity for fast diffusion equations in bounded domains

We will show optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is joint work with Jingang Xiong.

On the asymptotic optimality of the comb strategy for prediction with expert advice

For the problem of prediction with expert advice in the adversarial setting with geometric stopping, we compute the exact leading order expansion for the long time behavior of the value function using techniques from stochastic analysis and PDEs. Then, we use this expansion to prove that as conjectured in Gravin, Peres and Sivan the comb strategies are indeed asymptotically optimal for the adversary in the case of 4 experts.

For questions, contact Cornelia Mihaila at: cmihaila [at] math [dot] uchicago [dot] edu.